A study of the heat flow for closed curves with applications to geodesics

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O T T

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P H I U ( C ( <

•

A STUDY OF THE HEAT FLOW FOR CLOSED CURVES WITH APPLICATIONS TC GEODESICS

S.K. OTTARSSCN

July, 1986 Mathematics Institute

University of Warwick COVENTRY CV4 7AL

INTRODUCTION

Each of the two parts of this thesis is a discussion on questions arising from the 1964 paper by J. Eells and J.H. Sampson "Harmonic mappings of Riemannian manifolds" [ES]. To start with, here is a summary of the relevant material from that paper:

Let M (resp. M ‘) be a compact (resp. a complete) Riemannian manifold without boundary. The central problem in [ES] is that of deforming a given mapping f:M -*■ M ‘ into an extremal of the energy functional E, i.e. into an harmonic mapping. The method used, was that of proving (under certain metric and curvature assumptions on M') the existence of a solution f : H + H', s 6 [0,°°), of the heat equation, which coincides initially

(i.e. for s = 0) with the given map f, and then proving (under further assumptions on M ' , in the case where it is non-compact) that such a solution does in fact lead to an extremal. In more detail the proofs go as follows: first, by using a suitable embedding w of M' in some Euclidean space and constructing a Riemannian metric on a Euclidean tubular neighbour­ hood N of the image such that w : M’ -*■ N becomes an isometric embedding, the harmonic map equation and the heat equation were replaced by global equations (Eqs. (1) and (2) p. 140 in [ES], see eqs. (6.1) and (6.2) in Part I, eqs. (6.1) and (6.2) in Part II). Then, using these equivalent global equations, uniqueness and the existence of a solution f for small values of the deformation parameter swere proved (Section 10 in [ES]).

The length of the interval of existence was found to depend on the energy density of f and on a compact set containing f (Theorem 10(B) in [ES]). To prove existence for all positive values of the deformation

parameter it was therefore necessary to find a priori estimates for the energy density of the solution.

To prove the a.priori estimates for the energy density (Theorem 9(B) in [ES]) it was necessary to impose the condition that the target manifold had non-positive Riemannian curvature. To prove the a priori estimates for the second order space derivatives of wof$ (Theorem 9(C) in [ES]) some additional conditions, this time on the embedding w were imposed (conditions (12), Section 8(D) in [ES]). The first of these conditions, i.e. those on the covariant derivative of the projection map 7T:N -*■ M' because its coefficients appear in the global equations. Th e second part of (12) ensures that one can use the energy density as a measure on the first order space derivatives of the map w°fs - The conditions (12) are automatically satisfied when M' is compact. Under the above-mentioned conditions it was then possible to prove the existence of a unique solution of the heat equation of the kind required, for all positive values of the deformation parameter (Theorem 10(C) in [ES]).

After this result is established, the question remains: will such a solution lead to an extremal of the energy functional? As is pointed out in [ES] (Section 10(D)),for some non-compact manifolds M' there are solutions of the heat equation which are unbounded (i.e. solutions that leave every compact subset of M ' for large enough values of the deformation parameter.) Eells and Sampson found a new condition on M ', which ensures that this does not happen (Theorem 10(D)). This new condition is again given in terms of the embedding w and the covariant derivative of the projection map it.

Finally, Eel Is and Sampson proved (Theorem 11 A) that given a bounded solution fs , s e [0,°°), of the heat equation, if M' has non­ positive Riemannian curvature, then there exists a sequence s i>s2,s3>*** of s-values such that the mappings f converge uniformly, along with

sk

their first order space derivatives to an harmonic mapping. Here the curvature condition was used both to have thea priori bounds on the space derivatives of the solution and to ensure that the s-derivative of w°fs converged in the mean to zero as s - 00.

Now for a description of the material presented in this thesis. Throughout it is concerned with closed curves i.e. the domain M above is replaced by the unit circle S 1.

Part I is concerned with the problem of carrying out the method described above in the case of domain s \ without imposing any curvature restrictions on the target manifold M, and without any conditions depending on an embedding of the manifold. This problem is approached from an angle slightly different from that of Eells and Sampson as follows: first, for a given closed curve f, a condition on M will be given that will ensure that any solution f of the heat equation, which is continuous along with its first -derivative and which coincides with f at s = 0 has its image contained in a fixed compact subset of M. This condition is different from the one in [ES] and does not depend on an embedding. Then, assuming that that is the case (i.e. all solutions uniformly bounded) the existence of a unique solution defined for all positive values of the deformation parameter will be proved. The proof of this result will follow the corresponding proof in [ES] but the conditions on M (in particular the

curvature restriction) used there will not be necessary. Finally, in Part I, it is proved that this solution subconverges to a closed geodesic, again following [ES] but without the curvature restriction.

The assertion that the results of [ES] hold for closed curves on compact Riemannian manifolds without curvature restrictions was made in "Variational theory in fibre bundles" by J. Eel 1 s and J.H. Sampson, Proc. US-Japan Sem. Diff. Geo. Kyoto (1965) 22-23 and in "On harmonic mappings" by J.H. Sampson, Istituto Nazionale di Alta Matematica Francesco Severi Symposia Matematica Vol. XXVI (1982). The latter contains some remarks about the proof. The proof in this case (i.e. for closed curves on compact Riemannian manifolds of arbitrary curvature) formed the author's M.Sc. dissertation written at Warwick University in the academic year

1981-82. Part I appeared in the Journal of Geometry and Physics Vol. 2, n.1, 1985 under the title "Closed geodesics on Riemannian manifolds via the heat flow".

Part II is taken up with a discussion about the question :

What happens if one considers Lorentz manifolds instead of Riemannian ones? More specifically, given a closed curve f on a Lorentz manifold M, can one prove results about existence, convergence etc. of a solution f of the heat equation with fQ = f, of the same kind as those in [ES]? The change from a positive definite metric to a non-definite one turned out to change the properties of the solutions of the heat equation as is illustrated by some examples in Section 5 of Part II. For instance, there are examples of solutions for which the energy is not bounded and examples of solutions which only exist up to a finite value of the

deformation parameter. However, there are also examples of solutions which exist for all s 2 0,are bounded along with their derivatives and which

converge uniformly to a closed geodesic. Part II is then devoted to the investigation of to what extent one can carry out the method from [ES] described above. In more detail as follows: The idea that one could apply the results already obtained in Part I lead, to the study of certain Riemannian metrics naturally associated to time-orientable Lorentz manifolds. However, the result was that this approach only works in the special case where the Lorentz manifold has a parallel timelike vector field. As to the causal character of curves, it turned out that applying heat flow to closed timelike curves, the solution will preserve that property, i.e. will define a t-homotopy of the initial curve. The property of being a spacelike or lightlike closed curve is, however, in general, not preserved.

The simple property of solutions on Riemannian manifolds that the energy of the curves f decreases as s increases does not have an analogue on Lorentz manifolds (see examples III and IV of Section 5, Part II). For timelike solutions however, one can say a bit more both about the evolution of the energy and of the "length" as defined by the Lorentz metric, in fact, the length of the curves f for such solutions increases as s increases.

As for the existence of solutions, the main difficulty lies in the fact that one cannot use the energy density, in the same way as in [ES] and Part I , as a measure on the first derivative of the curves fs> both because the energy density is, in general, not bounded, and even if it were, that would, because the metric is non-definite, not imply boundedness

in any Riemannian metric. The proofs of uniqueness and of existence for small values of the deformation parameter are the same as for Riemannian

metrics, but in order to prove existence for all positive values it was necessary to make certain new boundedness assumptions on the first 9- derivative of the solution.

Finally, in Part II there is a proof of subconvergence of bounded timelike solutions with bounded first 9-derivative to a closed geodesic.

This thesis has benefitted from the valuable advice and guidance of Professor J. Eel Is.

notation used is fixed in Section 2 which also contains some basic definitions and results in differential geometry. The definitions of energy, tension field etc. are given in Section 3, along with some fundamental properties of solutions of the heat equation. The condition for boundedness of solutions is given in Section 4 and Section 5 has some results about when, in terms of the geometry of the manifold, such a condition might be fulfilled. The proof of existence for all positive values of the deformation parameter is in Section 6 and the proof of sub­ convergence of the solution to a closed geodesic is in Section 7.

2. NOTATION

Throughout M will denote a complete Riemannian manifold. <u,v> is the inner product of two tangent vectors u, v at the same point on M and [v| = <v,v>^ the length of v. d is the distance function on M and Br(x) the open ball centred at x with radius r.

The unit circle S 1 will always be parametrized by the central angle 9. If f is a mapping with domain S 1 x I where l i s a subset of F, for a fixed t in I ffc is the mapping with domain S1 given by ft (e) = f(e,t). f:S •» F will sometimes be identified with f » (9 ■+■ e ) :F F. a denotes differentiation with respect to 0, 5 f = f a where f is the differential_{0 * 0 *} of f, similarly for at .

The symbol V will be used for the Levi-Civita connection on M and for the induced connection on the vector fields along a smooth mapping f into M. The following facts about the connection (from [GKM]) will be needed.

Let X, Y, Z be vector fields on M. The torsion tensor T defined by

T(X,Y) = VXY - 7yX - [X,Y] (2.1)

satisfies T = 0. The curvature tensor R is defined by

R(X,Y)Z = Vx Vy Z - VYVX Z - V[XtY]Z.

Let N be a smooth manifold, f:N ->■ M smooth, A, B vector fields on N and X, Y vector fields along f. Then

0 = T(f*A,f*B) = V.f*B - V„f*A - f*[A,B]

R(f*A.f.B)X = VAVBX - VBVAX - V [A>B]X

3. THE HEAT EQUATION Definitions:

For a C 1 curve f:S1 •+ M its energy density e(f) is the function S 1 ■* F defined by e(f)(g) = i|3_f(9)|2 . The energy of f E(f) is

r2-n- 9 .

defined by E(f) = e(f)(9)de. The length of a C curve g L(g) is JO

the integral of the length of its tangent vector over its domain. 2

When f is C its tension field t(f) is the vector field along f given by x(f) = V3 30f . f is a geodesic iff its tension field vanishes.

9

Lemma 3 A :

Let fj.:S^ -*• M be a smooth family of closed curves for t in some open interval. Put E(t) = E(ft). Then

(2-rr

3tE(t) * - |

(9),3tft(9)>d6.

E(t) = J0

7<39^t(0),a0ft(e)>de

Further 0 = •2ir ■ 0 39O t ft(e),39ft(9)>de = [<Vg9 + O tft(9), 3tft(0),30ft (0)> V 3 39ft (9)>]d0>₉ and from this

■2ir

<va 3t f t (9)»3ef t (9)>de = '

0 0 <V3 39ft (9)’3tft(9)>d0 9

□

Corollary:

If ft is a smooth solution of the heat equation

then

3tft x(ft) (3.1)

(3.2)

Therefore 3tE(t) s 0 with 3tE(t) = 0 only when f^ is a geodesic.

The following propositions are special cases of Propositions 2(B) and 6(B) of [ES].

Proposition:

2 1

Proposition:

If (0,t) -*• -ft (9) is a map of S 1 x (tg,t^) ■* M which is C 1 on the

2 1

product manifold and C on S for each t, and if that map satisfies (3.1), then it is smooth.

4. BOUNDEDNESS CONDITIONS

Solutions of the heat equation that are not bounded exist as is shown by the following example from [ES].

Let N be the manifold obtained by revolving the graph of

v(u) = 1 + e u around the u-axis. If f satisfies 3 u = 0, <j> = 9 (<{> revolution angle) then so does the solution f^ for any subsequent

u .

time. The heat equation reduces to 3..U = % -■■■- . Thus eu + u - 2 log(eu+1) =

z e^u +1

= t + const, in particular u -<■ «> as t However, if the length of f is less than 2tt one would expect the solution to be bounded.

Fix a C1 curve f:S1 -► M. Throughout this section ft :S1 + H is a solution of the heat equation for 0 s t < b s =° which is continuous along with 3gft at t = 0 and which coincides with f at t = 0. As before E(t) = E(ft ).

Definition:

Let c > 0 and U be an open set in M. U has the property P(c) if

2 1

for every C curve g:S -*• U

cE(g) s (4.1)

Put m(f,c) » (2tt)~^E(f ) + (3ttJ + 2(2tt)"J )E(f )* + 4(2tt)_ic ~ 1 E(f)1/4.

Theorem 4A:

Let M satisfy the following condition: There is a compact K c M such that for every

x

of some f. lies in NSK then there exists a p £ M such that lim f. = p.

xo t-*~

Proof:

For every closed curve g:S^ ■» M one has the inequalities

diam g(S1) s j L(g) s TrJE(g)^ (4.2)

By the corollary to Lemma 3A E(t) is a non-increasing function and is therefore bounded by E(f). Bearing that and (4.2) in mind it is seen that if all the curves f^ with t £ (o,b) intersect K then the image of the solution is bounded.

So suppose that the image of ft lies in I'KK for some tQ £ (0,b). Fix 0Q £ S1. By hypothesis there exist c > 0 and k > m(f,c) such that Bk (ft (eQ )) has the property P(c). It is easily seen that

o

ft (S1) e Bk (ft (9Q )). Put t1 = sup {t ’ > tQ :ft (S1) c Bk (ft (eo )) for

*“0 O 0

all t € CtQ ,t *)> and suppose for a contradiction that t^ < b. It is easy to show that

sup d(ft (e),ft (0)) s TTi[E(t0 )i ♦ E(t, )JD

0£S1 1 0

(4.3)

1

THFJ

2IT .*1

f

|3f f t (e) |dtde £ (2ir)-i ^

(

Jt«

jt

0

J

t0

Put

T, - {t i

* 1},

=

>

£

T3 = {t 2 tQ : -3tE(t) £ E(t)*}. From (4.4)

1 7n |3tft(0)|dtde £ (2tt)_J ♦ (2-rr)-i f J[t0 ,ti]nT2 t ))^dt + J c W ftTi (-3i.E(t))idt+(2TT)'i f 1 I

ii

(-3tE(t))*dt £ (2ir) ” ^ (-3fE(t))dt

Ct0»ti^nTl

(2ir)"*(E(tQ ) - E(t,))

2 i.e.

E(t

-3t E(t)

t € [ t ' , t

3tE(t) 1 5 I-E( t ) * t" - t'£ -2 - -23t(Ei) t" i 3t (E*)dt

t'

z

].

Therefore, the sum of the lengths of the intervals making up [tQ ,t^] n T2

III Since Bk(ft (eQ )) has the property P(c) one has

and therefore 3tE(t)

--- £ -c, for all t a [t„,t.).

E(t) 0 1

From this it follows that for t € [t ,t^) is no greater than 2(E(tQ )^ - E(t1)^) and so

(••¡^(t^dt < 2(27r)*4(E(t0 )i - Eit, )4 ) (4.7)

0

cE(t) £

E(t) £ E(t0 )e'c(t'to ) (4.8)

Therefore (-3tE(t))idt £ (2ir)_i

j

sup d(ft (e),ft (0)) < irJ[E(t )* ♦ E(t, )*3 + (2ir)_i[E(t ) - E ( t J ] +

0£S1 1 0

+ 2(21r)'i[E(t0 )i - E U , ) * ] ♦ 4(2Tr)'ic ‘1E(t0 )1/4( 1 - e x p [ - ^ - ^ - ] ) . (4.10) And further

sup d(ft (0),ft (0)) S (2Tr)-iE(t0 ) + (27Ti+2(27r)’i)E(t0 )i+4(2Tr)‘ic' 1E(t0 )1/4.

0€S1 1 0

From this it follows that f. (S1) c Bk(ft (0 )), but that would mean

L1 K Lo 0

that there is an e > 0 such that f fc +^(S1) <= Bk(ft (0Q )) for all t £ [0,e) contradicting the choice of t,.

For the proof of the last assertion of the theorem observe that by the above proof ft(S4) <= Bk (ft (0Q )) for all t £ [t0 ,=>) and therefore the inequality (4.8) will hold for all t £ [t0 ,°°). Further, for all t',t" £ Ct0>=°) the inequality (4.10) will hold for t 1, t" in place of t , t.. For a fixed 0 (4.8) and (4.10) show that ft (0) converges to a point p as t -*■ 00. Further, (4.8) shows that E(t) and therefore (by (4.2)) L ( f J converge to zero, which shows that lim f (0) is independent of 0.

t-*o° Remark:

In the case of M = F n it is easy to show that the inequality 2tt

E(f) <

f

I

I

Jo

First consider real valued f. For any f £ L^(S^) the Fourier coefficients of f are defined by the formula

n € Z

and one has the Parseval theorem

I f(n)g(n) = ^ n=-°°

f(e)gT9)de

2 1

whenever f, g e L (S ); the series on the left converges absolutely.

2

When f is C one has

30f(n) = inf(n), a2f(n) = -n2f(n) n € Z i fir ? 00 ^ o 00 9 J- la0f(e)| de = Z |30f ( n ) r = Z n^|f(n) '-it n=-°° n=-o° J - f |a0f(e)|2de = Z n4 |f(n)|2 . J-TT 9 n=-=°

Therefore ||a0f||| S||a|f || | and the same inequality holds for R n-valued f because it holds for each component.

5. MORE ON BOUNDEDNESS CONDITIONS

In this section a demonstration of the inequality

\ E(f) < f2Tr|x(f)12de

2tt

Jo

for closed curves satisfying certain assumptions, following a proof of the Synge-formula given on pp. 122-3 in [GKM].

Definition:

If A is a subset of M define

k a = sup {<R(X,Y)Y,X> : X.Y € T M , |X| = |Y| = 1, p € A}.

Then for all p € A, X.Y € TpM, <R(X,Y)Y,X> s <A |X|2 |Y|2 .

Theorem 5A:

1 2

Let f:S ■+ M be a non-constant closed C curve and assume w.l.o.g. that the maximum of the energy density occurs at 9 = 0. Suppose that the image of f is contained in a ball Br(f (0)) satisfying the following: The exponential map exPf(g):Tf(ojM ->■ M restricted to Br(Q) in T ^ g j M is a diffeomorphism onto B_(f(0)) and if <„ > 0 the radius r satisfies

r d 2^

r < I2icr)’*. Then the inequality, — U- E(f) £ I It(f ) i2d0 holds.

b

2n

J

0

Remark:

The inequality is of course trivial when f is constant.

first e > 0 such that f(e) = f(0). Put B = Br (f(0)). To begin with a few definitions.

1 _i

Define f*:S -*• T ^ g ^ M by f* = ex p ^ g j n f and a family of geodesics g by g(9,t) = e x p ^ g ^ (t .f *(0)). For a fixed 9 g9 (.) = g(9,-) is a geodesic from f(0) to f (9) (g9 (0) = f(0), g8 (1) = f(g)).

Define three vector fields along the map g: X =3t96 » Y = 30g9 and Y - Y - < Y *X>X/|X|2 .

Define L (9) to be the length of g9 . L (0) = |atg9 | = ¡X|.

Step I:

lim X/|X|(0,t ) exists and equals the unit vector in the direction 0H-O +

Proof:

Identify (gjM and all its tangent spaces with F n and so let Jy (u) be the vector u translated to One has

W (6,t) = CXPf(0)* (jtf*(9) - {'(g ) [ ]

(compare Gauß-lemma p. 136 [GKM]). lim — f.JJJ. exists (in Tjr/nxM) and 9-*-0+ |f*(0)|

equals the unit vector in the direction of s.fl n and so lim -r-2-y (0,t)

0 9=U g^Q+ l*l

exists and is equal to the same vector. □

Since g(6,0) = f(0), Y(0,O) = 0 and also V, Y(0,O) = 0 •,

39 j (5.1)

Since g(0,1) = f(e), Y(9,1) = 3„f(0) and also V. Y(0.1)-x(f)(0)

The vector field Y along g (the component of Y orthogonal to X) is C 1 for 9 £ (0,9o ) and Y(9,0) = 0 by (5.1).

Step II:

3gL(0) = Og f( 9) i jyy (0,1)> (5.2)

Proof:

By (2.2) V. X = V. Y . (5.3)

30 3t

Using this and L(0) = [ |X| dt there follows JO

30

L(9) =

_W

W

This follows immediately from (5.2) and Step I. o

Step III:

302L(9) = yjq- « V g tY,V3J > - <R(X,Y)Y,X>) dt + X

• w

Proof:

By (5.3) 3e (-pq-) ■ 30<X,X>_i = - <X,X>"3/2<73 X,X> =

<7, Y,X>

|X|

f

1

1

From above 3QL(e) = -po-r <7. Y,X>dt, therefore using (5.6) and

e Jo I I 3t

one obtains

39

L(e) '

l

39

<

73

t,-X>)* <9e m><V - x»', t "

■J

o i x i 3e 3t 3t 3e |x| 3t

= f1 (4-rUv., v a Y *x> + <7, Y.V, Y» - — 4 <7, Y <

Jo 1 1

30 3t

3t

3t

|X|

3t

Since for a fixed 9 g (9,*) is a geodesic 7 X = 0 and 3^|X| = 0.

7. ( - 4 - <Y,X>X) = — U (3*<Y,X>)X

3t I X| ^ | x p r

Also, since <Y,X> = 0

<7a Y,X> = 3t<Y,X> = 0

Now Y = Y + — 4 <Y,X>X and 7, Y * 7. Y + 7, (— 4 <Y,X>X)

| X P t 3t 3t [ X p

1

7. Y + — (3.< Y,X»X by (5.8). From this it follows that 3t | X p z (5.6) (5.3) X>2 )dt = X>2 )dt. (5.7) Therefore (5.8) (5.9)

<V. Y,V Y> = <V Y,V. Y> + at 3t 3t 3t 1 |X|‘ (3i.<Y,X>) .2<V. Y,X> + + — Ur 0 1.<Y,X>)2 <X,X>.

ur

1

The second term in this expression vanishes by (5.9), the last one equals — i-rrKv. Y,X>)2 . In other words

|X|2 3t

<V. Y,V. Y> = <7. Y,V. Y>) + — L - (<V„ Y,X>)2 .

3t 3t 3t 3t |X|2 "t

Substituting this in (5.7) gives

seu e ) ’ ! ’ W (<,ae\ v ’x> * < V ' V > w t

By (2.3) V V Y = V. V Y - R(X,Y)Y. Therefore (V. V, Y,X> =

On Of 0*. aa df.

= <7, V. Y,X> - <R(X,Y)Y,X> = 3t (V. Y,X> - <R(X,Y)Y,X>.

3t 3e

z

(5.10)

Substituting this in (5.10) gives

32L(0) = (3t<V3 Y, y£r> + - ^ ( < V 3tY,V3J > + <R(X , Y) Y ,X>) )dt. (5.11)

From p. 91 [GKM] <R(X,Y)Z,U> = -<R(X,Y)U,Z> and therefore <R(X,Y)Z,Z> = 0. One has also R(X,Y)Z = -R(Y,X)Z and therefore R(X,X)Z = 0.

Put h(e,t) <Y,X>. Then Y = Y - h.X.

<R(Y,X)X,Y> = <R(Y,X)X,Y> - <R(h.X,X)X,Y> =

= <R(Y,X)X,Y> - <R(Y,X)X,h.X> = <R(Y,X)X,Y>.

This shows that <R(X,Y)Y,X> = <R(X,Y)Y,X>. This changes (5.11) into

which is (5.5). o

Step IV:

The integral I = [ j L « 7 , Y,7, Y> - <R(X,Y)Y,X>)dt

JO 3t 3t

is non-negative.

I = t4t i « V , Y,7. Y> - <R(X,Y)Y,X>)dt since !X| is independent lA l Jo 3t 3t

3?L(0) = f1 — (<V. Y ,7. Y> - <R(X,Y) Y , X » d t + ['

9 JO | X | 3t 3t JO

3

t<vaeY*

7

*T>dt =

Proof:

of t.

Obviously if <B s 0 the integral is non-negative.

~,2

(e,t) = <Y,YXe,t) - <Y,Y>(e,0) = [ 3.<Y,Y>dt = f 2<V Y,Y>dt <

JO z

Jo

3t

2 f V . Y ||Y|dt < 2( f1 |v, Y|2dt)J(

Jo dt Jo dt

|Y|2dt)J .

Therefore,

f1 |Y|2dt < 2(f1 17. Y|2dt)i( [1 |Y■ 2dt)“

JO JO 9t JO

Or

[1 |Y|2dt < 4 [1 |7

JO Jo 3t

Y ] 2dt

Since the radius r of B satisfies r<(2i<g)"* one has |X| s (2iCg)”^, Therefore

<R(X,Y)Y,X> S <b|X|2 |Y|2 s j |Y[2 . Consequently

<R(X,Y)Y,X>dt s l f1 |Y|2dt s |y Yf

H Jo

Jo dt

(5.12)

using (5.12). This gives

f

1

- ?

- ~

(IV_ Y|£ - <R(X,Y)Y,X>)dt a 0. □

Jo

dt

By the corollary of Step II lim a L(e) = |3 f(0)|. From this one

9-0+ 9 9

30L(e) = |30f(0)| + 3^L(01)de'.

0 9 From Steps III and IVthere follows

30L(9) >

Since f is a closed curve there must come a 91 where 30L(6j) = 0.

By assumption the energy density attains its maximum at 0 = 0 so

To conclude this section two examples of how one can combine the results of this and the preceding section.

Define i :M -*■ F + by

i(p) = sup {r:exppTpM ■* M restricted to Bp (0) c Tp M is a diffeomorphism onto Br(p )>.

Then

Case 1:

There exists a compact K c M such that k^ k S 0 and i[ M^ K£ 2rQ > 0. Then any solution of the heat equation with initial curve f such that mif.-ly) < r has bounded image.

2ir 0

Proof:

For any p such that B, (p) lies wholly in NKK, jro

1 2

property P(— 7 ). For let g be a closed C curve with

2

ir

assume that the maximum of e(g) is attained at 9 = 0. in NKK and so satisfies the hypotheses of Theorem 5A. applies and the result follows.

Similarly: Br (p) has the

o

0

There exists a compact 1 ¡NKK ’ 2ro * (2< M^K ^ * Then curve f such that m(f,— <

2

ir

K <= M such that k,. K > 0 any solution of the heat rQ has bounded image.

and

6. EXISTENCE OF SOLUTIONS OF THE HEAT EQUATION

This section will be devoted to the following existence problem: Given a closed curve on M does there exist a solution of the heat equation defined for t € [0,t1),t1 £ °° , such that f^ and 3 f are continuous at t = 0 and f. coincides with the given curve at t = 0?

This problem will be treated under the assumption that the manifold M satisfies a condition (for example one from the previous sections) which will ensure that any solution of the kind sought will have its image contained in a fixed compact set. A solution of the heat equation will henceforth mean a solution of this problem for a fixed but arbitrary fQ .

The main ideas of this treatment are those of [ES]; therefore only an outline will be given with what is different here pointed out.

I. In [ES] it is shown how to replace the harmonic map equation t (f ) = 0 and the heat equation 3tft = t(ft ) (which in terms of local

coordinates on M are local systems of equations) with global systems. This is done as follows: M can be smoothly and properly embedded in some Euclidean space F q by a map w:M -»F^. Given such an embedding it is always possible to construct a smooth Riemannian metric on a tubular neighbourhood N of M so that N is Riemannian fibred. Let tt:N -► M be the projection map and "Its covariant differential. Then

(a) A map f:S1 ■* M satisfies i(f) = 0 if and only if the composition W = w°f satisfies

(b) A deformation ft :S -*■ M (tQ < t < tj ) satisfies t(ft ) = 3 ^ if and only if W t = w°f^_ satisfies

se"Ì - 3t “‘t ■ "ai»5s“at39“Ì

Also, given a smooth Wt :S1 -»• N satisfying (6.2) for t s t < t 1 , if maps S1 into M then so does every W. for t < t < t..

to

z

0

II. Derivative bounds Lemma :

Any solution ft :S + M of the heat equation has energy density satisfying

3ee < V - \ e < V - l ^ ft )l2 (6.3)

Proof:

39

*<V ■ ^<rVt.

3

eV> ’

3a < V e W t > ■ <73e7393ef t ’3e V *

The heat equation can be written aQft so

0

3t e(ft ) = <7

t 3ef t ,8

f t > * <7303t f t*30f t > = <73973930f f 30 V

39e(ft ) - 3te(ft } = <V3A30ff 73o30ft> = _{0 0} °

, . 1 - ( e , - 0 ? ) 2

Put H(e1 ,e2 »t) = J (nt) 5 exp [-- ^ - = — ] for G F , t 6 F +

H is a fundamental solution for the operator L = af - _{0 0} 3 , satisfies_t L„ H = La H = 0, and the identity

e 1 32

u(e1ft) = - dT H(01t82 ,t-T)LQ u(92 ,T)de2

♦ | H(01,02 ,t-to )u(02 ,to )de2 t0 < t < t1

holds for all u t defined on S which areof class C in 0 and C in t V t s v

Suppose f^ is a solution of the heat equation defined for 0 £ 1 Since H > 0 there follows from (6.3) and (6.4)

e(ft)(01 ) £

j

For t > 1, putting t-1 for tQ in (6.5) there follows

e(ft)(01 ) £ j H(01,02 ,1)e(ft.1)(02 )d02 and therefore

f2lT

e(ft) £ const

J

(6.4)

for

< t^.

Any smaller value can be put in for t-1 on the right for example zero since e(ft ) is assumed to be continuous at t = 0.

For 0 < t < 1 put t = 0 in (6.5) to obtain

e(ft )(e1) < ; H(e1,e2,t)e(fo )(92)d62

Put e(f ) = sup e(f )(e) then (_{O * 0} 6.6) shows that

ees

e(ft )(e) £ const. e(fQ ).

(6.6

To summarize:

Theorem 6A

Let f^ be a solution of the heat equation for 0 i t < t|. f2ir e(ft ) s const. e(ft ) s const. e(f J( e ) d e 0 0 sup e (f )(e ) 1 0 ees' for 1 $ t < t1 for 0 S t S 1 Then

with the constants not depending on ft .

The difference between this and the derivation of bounds for the first order space derivatives in [ES] is that the curvature terms in the identity (6) §8A in [ES] do not appear in the corresponding identity here (i.e. (6.3)), because the domain is 1-dimensional. It is therefore not necessary to impose curvature restrictions on the target manifold M.

For a solution of the heat equation there is the following theorem, proved in [ES], for the second derivative with respect to 9 of the Wt in (6.2).

Theorem 6B

Given z there is a constant C independent of t such that |3gW£| < C for t > e. The constant C depends on fQ .

This is proved by using the formula

fundamental solution H. The only remark that should be made is that here it is assumed that the manifold satisfies conditions which will ensure that the image of any solution will be contained in a fixed compact set and therefore the embedding conditions in [ES] are not necessary since the inequalities (12), §8D in [ES] are automatically satisfied on a compact set.

Ill The following two theorems are proved in §10 [ES].

Theorem 6C

If ft and f^ are two solutions of the heat equation with fQ = f^ then they coincide for all relevant t > 0.

where

Theorem 6D

Let M ‘ be a compact subset of M. Then for any closed curve fQ :S^ -*• M such thatfjs') lies in M1 there is a positive constant t^ depending only on M‘ and the magnitude of the energy density e(fQ ) such that there exists a solution f. for that f for 0 < t < t..

t o l

From these two theorems one deduces the following which corresponds to theorem IOC in [ES].

Theorem 6E

There is a unique solution ft of the heat equation defined for all t 2 0.

Proof

Such a solution exists for small t by Theorem 6D and is unique by Theorem 6C. Let t1 be the largest number such that a solution of the kind sought exists for 0 s t < t ( and suppose that t1 is finite. By assumption the manifold M satisfies conditions which ensure that the images ft(S1) (0 2 t < t.) all lie in a compact subset of M. Theorem 6A shows that the energy density remains bounded and therefore by Theorems 6C and 6D there is a fixed positive number c such that any ft can be continued as a solution into the interval (t,t+e). This contradicts the finiteness

7. SUBCONVERGENCE OF SOLUTIONS

In this section let f be a fixed closed C curve S ->■ M and assume o

that M satisfies conditions which will ensure that any solution of the heat equation f^ which is continuous along with 3aft at t = 0 and which coincides with the given f at t = 0 will have its image contained in a fixed compact set. Then by the preceding section a unique solution f^ exists for all t € [0,~). In this section a proof of the following:

Theorem 7A

There is a sequence t^.tg.t^,... with tk -* °° such that the curves f. = f. converge uniformly to a closed geodesic f.

Lemma (See [H])

For k (ft) = ^■Otft ,Stft> one has the fo11owin9 identity

(7.1)

Proof

■ <va V, 3* f,,3tO_{3g O g V ' t ' V f} + <V. 3*f*.V- 3 , 0 ■ 30 t* “9 t t

3t C? <3tft ,atft>] = <V3t3tft ,3tft>

= <V

3tV3039ft'3tft>

Combining these it is easily seen that

' V i . V t ' V t ' " aecI<st f t ’ 3t f t >]

which is (7.1). □

Lemma

3tE (t ) + 0 as t + ®. (As before E(t) = E(-ft )).

Proof

rt7t

Put K(ft ) = j k(ft)(e)de with k(f^) as in preceding 1 3tE(t) = -2K(ft ) (Corollary of Lemma 3A) and

r2n

emma.

3‘E(t) = -2 3tk(ft )(e)de.

The last term in (7.1) is uniformly bounded for t greater than any given positive number by Theorems 6A and 6B. By integrating (7.1) over it therefore follows that there exists a constant C such that 3^E(t ) > C. 3tE(t ) cannot be bounded away from zero because it is integrable so if C 2 0 then obviously 3tE(t) -► 0. Suppose C < 0. If for some e > 0 there exist arbitrarily large t such that 3tE(tQ ) = - e then

t o

( - e -C(tQ -t)) = e2/2C 3tE(t)dt S

V e/C

but this contradicts the integrability of 3t E(t). □

Rema rk

This lemma will be used in the proof of Theorem 7A. In [ES] the corresponding result (Corollary § 6 ( 0 ) is proved with the assumption that the target manifold has non-positive sectional curvature. This assumption is not necessary here.

In the following proof it will be convenient to use the function G defined by

where a bj is the characteristic function of the interval (a,b). The formula

f2lT

2ir

h (

9

) = h (

) -

3

a h (

0

) (

2

i r - e ) - f

G ( e , e ' )

3

? h (

0

' ) d e

holds for al 1 C2 functions h:S1 -<■ R and

0

(7.2)

Proof of Theorem 7A

Let W t = w°ft be the solution of (6.2) which corresponds to The mappings and 3gW t form bounded equicontinuous families (Theorems 6A and 6B). Therefore there exists a sequence t^.tg.t^,... with tk -» °° such that the mappings W, = W.. converge uniformly along with _K 5 R to_{0 K} continuously differentiable mapping W. The Wk can be represented by the formula r2ir wk(9) = wk(0) ' 39wk(0)(2lT-9) * J Gie.e’JSgW^e'Jde1 or r2tt Wk(0) = w‘ (0) - 30wJ ( O )( 2tt- 0) - J o G(e,9' ) ( Fk(e' ) ♦ 3t Mj(0*

where f£ = tr^b 3QWa 30Wb . By the preceding lemma the 3tw£(0) converge in the mean to zero as k + ». Therefore, since G is bounded

f2ir

-lim I Gte.e'^jrie'Jde' = 0.

k « iO 1 K

Passing to the limit in (7.3) there results for the mapping W

W c (9) = WcL0)- 39Wc(0)(2tt-9) -

•2

tt

G(0,9‘)Fc(0,)d0

where Fc (0') = lim F^(0*) = ^ ( W j s Wa 3QWb . Referring to (7.2) it is k-KOO

seen that M satisfies (6.1) which means that it corresponds to a closed geodesic. a

) )d0'

PART II 1. CONTENTS

The material of this part is arranged in sections as follows: after some preliminary material in Section 2, Section 3 is about the decomposition of Lorentz vector spaces associated to timelike vectors and about Riemannian metrics associated to a time-orientable Lorentz manifold. Section 4 is about the evolution of the energy and length of solution curves while Section 5 is devoted to examples. Sections 6, 7 and 8 contain the discussion about the existence and uniqueness of solutions and Section 9 is about the subconvergence of timelike solutions.

2. PRELIMINARIES Definition

A Lorentz manifold (M,g) is a smooth manifcfld M of dimension m > 2 with a symmetric nondegenerate (0,2) tensor field g of constant index

v - 1.

The index v of a symmetric bilinear form b on a vector space V is the largest integer that is the dimension of a subspace W c V on which b|W is negative definite.

g(u,v) will usually be written <u,v>.

As for Riemannian manifolds one has the Levi-Civita connection associated to Lorentz manifolds and the same notation and properties as specified in Section 2 of Part I will be used. In particular the geodesic equation is written the same way

V3 3Qf = 0 (2.1)

9

Definition

A pregeodesic is a curve that has a reparametrization as a geodesic. The following result will be used ([O'N] pp. 69, 95-96).

Lemma 2A:

A curve f is a pregeodesic if and only if the tension field V0 30f 0 and the tangent field 30f are everywhere col linear.

Definitions

A tangent vector v to a Lorentz manifold (M,< , >) is spacelike if <v,v> > 0 or v = 0

lightlike or null if <v,v> = 0 and v ^ 0 timelike if <v,v> < 0.

The light cone at p is the set {v e TpM : v lightlike}

A curve a on M is spacelike if all of its velocity vectors are spacelike; similarly for timelike and lightlike.

On a time-orientable Lorentz manifold M [O'N pp.144-145] there exists a smooth timelike vector field y.

Definitions

Let f be a curve on the Lorentz manifold (M, < , >). The energy density e(f)(9) = i<30f(9),30f(0)>, the energy of f

E(f) = j 1 e(f)(0)de. When f is timelike define i,(f)(e) = ( - O0f (0),3Qf(e)>)* and

L(f

j

1 £(f)(e)de.

Unlike the Riemannian situation the quantities e(f) and E(f) can now take on all real values. The quantity L(f ) is unchanged by monotone reparametrizations £0'N p. 132].

3. RIEMANNIAN METRICS ASSOCIATED TO A TIME-ORIENTABLE LORENTZ MANIFOLD I. Lemma 3A [O'N p.141J : Let y be a timelike vector in a Lorentz vector space (V, < , >) (i.e. V is a vector space on which is defined a non­ degenerate symmetric bilinear form < , > of index 1). Then the subspace y1 (i.e. {v € V : <v,y> = 0}) is spacelike and V is the direct sum

o X

R*y + Y . q

Let (V, < , >) be a Lorentz vector space and y a timelike vector in V. For any vector X in V, its decomposition relative to y can be written

= 4 * 4 v + [X -_{< Y » Y > < Y *Y >} (3.1)

One has then

<X,X> <X,y>2 < Y .Y > + <X <X,y> ^7T7> Y > X <XT£> > < Y . Y > Y (3.2) where the first term on the right is negative and the second term positive. II. Let (M, < , >) be a time-orientable Lorentz manifold. Then there exists a timelike vector field y on M. One can assume that <y,y> = -1. Then one has (see [A])

Lemma 3B: < , defined by

<X,Y> = <X,Y> + 2<X,yXY,y> defines a Riemannian metric on M. □

It is easily seen that

and that for a timelike vector X

||X|| - (<X,X>Y )i < (2<X,Y>2 )i - / 7 | < X . y>|

and (3.3)

(-<x,x>)i * / 7 ;<x,y>|

Let vY denote the Levi-Civita connection associated to < , > . Proposition 3C

VY = V if and only if y is parallel i.e. the map X -<■ V y is X identically zero.

Proof

Since <y»y> = _1 one has

< V v , Y > = 0 (3.4)

X

for every tangent vector X.

Let X, Y, Z denote vector fields on M. It is easy to show that

<V^Y,Z> = <V V,Z> + [<Z,y> « Y , V y> + <x,v¥y>) +

X Y X Y X •

+ <Y,y> « Z , Vxy> - <X,VZY>) ♦ <X,Y>(<Z,VyY> * <Y,Vzy» ] (3.5) Clearly v = vY if and only if the expression in the bracket in (3 .5 ) vanishes for all X, Y, Z. Therefore it is immediate that y parallel implies V * VY .

Conversely, suppose that the expression in the bracket vanishes for all X, Y, Z. (This is a pointwise condition, the expression is linear in X, Y and Z, it therefore suffices to consider tangent vectors at each point).

Choosing Z = y and Y,X X y in (3.5) one sees that

<Y,Vxy> =-<X,7yY> for all X.Y X y .

Similarly choosing Y = y and X,Z X y in (3.5) one obtains

<Z,Vx y> = <X,Vz y> for all X,Z X y. The conclusion is that

<X,VyY> = 0 for all X,Y X y. (3.6)

Now, choosing Z X y. X = Y = ^ i" (3.5) there follows

<Z,V y> = 0 for all Z X y. (3.7)

4. THE HEAT EQUATION

The heat equation for closed curves on Loyentz manifolds is defined in the same way as for Riemannian manifolds i.e.

V s

Vs

This equation and the geodesic equation (2.1) have the same form as the corresponding Riemannian equations and will therefore have the same smoothness properties as quoted at the end of Section 3 of Part I.

I . Proposition 4A:

Let fg : S1 -*■ M be a solution of the heat equation for s in some 1

o " * ' s. b

interval I c F . If there exists an s 6 I s.t. f :S M is a timelike curve then the fg will also be timelike for all subsequent s € I.

Proof

Suppose that f : S1 -*■ M is timelike and that

max { O . f , , 3 af,> : 9 _{O S U S} 6 S 1} < 0 occurs at 9„._u Then W s ’ W 6=9, = 2<V 6fs * V s > ' 9=e, and (4.2) 36 < V s * V s >l6=9, _{2C<V V} 3-f- V s > + <W_{0'S 3e 9 s '}s*V 3_'9 oV s>]9 = 9 , £ 0 (4.3)

Further

3 c < V c * V c > l_{s' e s ’ 0 s''e=0} = 2<V3s39f,3efs>|_{e s '1e=e}

0 0

0 30 39°9,s ,oe's'le=e0 •

(4.4)

by Lemma 3A, and therefore by (4.3) that <V3 Vg 3Qfrs ,3Q-fs>|e_Q £ 0. By

The above Proposition shows that deformations by heat flow of timelike curves defines a t-homotopy i.e. a homotopy through timelike curves.

There is no corresponding result for the minima of the energy density of solutions (see Example IA in the next section).

The property of being spacelike is not preserved by the heat flow (Example IIIB, next section).

II. Putting E(s) = E(f$ ) one obtains in the same way as in the Riemannian case (Part I, Section 3) the formula

9 9 o ( 4 . 4 ) t h i s i m p l i e s t h a t 3cO a f <:»3flf,>! S 9 s’ 9 s '9=9_o S O . D Remarks

V 9 5 V

s

Because of the nondefinite nature of < , > this does not give any result analogous to the corollary of Section 3, Part I.

For the remainder of this section let f denote a timelike

solution of the heat equation. Decomposing 7, 3flf into factors parallel 30 0 S

and orthogonal to 30fs (Section 31) one can write (4.5) as

3sE(s) = r2ir

Jo

+ ! * ^3 V s ---- --- V s*V3 V s --- W

r

de

y s y s

zero). The first integral on the riqht of (4.6) is always non-negative and the second always non-positive. Using (4.6) one can prove

Proposition 4B

Suppose that each of the curves f : S^ - M is of constant energy density. Then 3sE(s) s 0 for all s.with 3gE(sQ ) = 0 for some sQ only if f is a closed geodesic.

5o Proof

By hypothesis <3afc »3af,> depends only on s. Therefore_{U S U S}

h <3.f ,3„f >

a 9 S ’ 9 S <739 39 fS*39 fS > E 0

(4.7)

(4.7) shows that the first integral in (4.6) vanishes for all s. Therefore 3eE(s) s 0 for all s. (4.7) also shows that 7, 3nf. is everywhere

space-s 3e 9 s

like. Therefore one can only have 3_E(s ) = 0 if

7.

. S 0 39 9 S0

W

(s) ■ " " t « 8e W

. » a»<1 W

s > ■

(<3e V 39f s >:

For the first integral in (4.6) one has the following estimate

Proposi tion ! « • * « < * » * <

-, 2 (*2

<va 3efs,3efs>

<- W a»*> a L -

d9

‘°

<39f s*38 V

III. Put L(s) = L(fs ) =

J

r 2tt 3sL(s)

. <*3 30fs ’30fs>

( - < V s > V s » ^ ^ 36f s ’73a V s > - <a

V

7 'f~ T ~ ]de

O 9 0 <30t S* 0 TS>

d9 □

Proof

3sL(s)

2tt 2tt

V - <3e V 3 *9f s>)id9 = - } oTr(- <39f s » 30f s> ) ‘ i < 7 3s30f s ‘ 38f s>de

J0 r2ir | 10 i-<39f s*3efs>) -i<V. V 3 f

39 °e 9 s

V f W ' V

« V e V W

(-<

9

e W

.>,3/2

'

Substituting this into (4.9) gives (4.8). □

Corollary 4D

3 L(s) a 0 for all s and } L(s ) = 0 only if f is a pregeodesic,

s S O Sq

Proof

Write V. 3.f = xi*(e) + xf(e) where _{0 S S S} t₅ 'Kq) (resp. xf(9)) is parallel₅

” i

(resp. orthogonal) to a f . x is then of course spacelike. The expression 0 s

inside the bracket in the integral on the right hand side of (4.8) equals <x1,x'L> Z 0. <x^ ,x^ > equals zero only when x^ * 0 which implies by

so so o

5. EXAMPLES

In this section a few examples are given of solutions of heat equations and their properties to illustrate some kinds of behaviour that can occur and some differences and similarities between the Lorentz and Riemannian situations.

I. Let M be the manifold F x S1 parametrized by r 6 R and the central angle i(i£ s'. Let h : R+iR\{0} be a smooth function. Define a

2 2 2

metric tensor g on M by ds = dr + h(r)d<j> . When h > 0 this gives a Riemannian metric on M, when h < 0 a Lorentz metric.

By using the formulas given in [O'N p. 80] one can calculate the Christoffel symbols of g. They are

One has

9l f W

= h ’ 912 = 921 = °* 922 = <

W

= 1

r = r = r = = 0

111

M 2

*22

l22

u

The heat equation can therefore be written

(5.1)

Suppose that the function pr satisfies o

3spr ^ = ‘ V ^pr

o o

I

for s in some maximal interval I containing 0.

Define f : S x I -*■ M by (e,s) (or (s),e). Then f is a solution o

of the heat equation (5.1). As is easy to see from (5.1) and (5.2) the

ment is not true for general solutions f; as is seen from (5.1) it depends on the property that 3 r = 0].

0

The following specific examples will illustrate some properties of solutions of the heat equation for Lorentz metrics and differences from the Riemannian situation.

A. Take h = ± er. The differential equations

map S x (-1) M defined by (e,s) -*■ f(e,-s) is a solution of the heat

2 2 2

equation for the metric ds' = dr - h(r)d<}> . [Remark: this last state­

o

o

< (

0

) =

r,

- P

p* (s) = - ln(e 0 ± |s) o

where p” (resD. pji ) is defined on I" » (-«>,2e 0

o o

) (resp.

o

-r

0 ± ^-s) ,6) is then = dr2 ± erdd>2 . For the 2

a solution of the heat equation for the metric ds± energy density one has

e(f* )(e,s) = j O g f * ,aQfJ X e . s ) = ± J(e 0 ± Js)'1

0 “ 0

This example shows that solutions of the heat equation for Lorentz metrics need not exist for all positive values of the deformation parameter s. Also, in this example one has for E(s) = E (f~ $ )

os

• p — p

3sE (s ) = - ^(e 0 - -^-s) < 0 and 3gE(s) = - ^-(e 0 - ^s) < 0.

B. Let M be the manifold R + x S with the Lorentz metric

2 2 a 2

ds = dr - r d$ ,a € R ^ {0}. The equation a a- 1

V r / Î T S [

»,„(0) - i*0

has the solution

(S> . {

1

(ro 2‘a t J a f Z - a l s ) ^ a t 2

roe a = 2

2r 2-a

For a < 0 the solution exists for s e Ia = ( - " , --- ) and

-2r 2-a a(2-a)

♦ D a s s + --- ---- and lim P (s) = ® .

a(2-a) s~-® ro

(For a = 0 the map (e,s) (r .8) is a solution of the heat equation), 2r 2-a

For 0 < a < 2 the solution exists for s 6 Is 5 (--- .«) and

Pr (s)-*® s , or (s) + 0 as s + o -2r 2-a a(2-a) as s -*• ® a ( 2 - a )

e(f* M e , s) = J o0f* ,a0fj X e . s ) = ± £(e i - ' ro . 0 ± ^1S)-1 0 v 0

This example shows that solutions of the heat equation for Lorentz metrics need not exist for all positive values of the deformation parameter s. Also, in this example one has for E(s) = E(f* )

o

_ p — p

3sE (s ) = - ^(e 0 - irS) < 0 and 3gE(s) = - ^(e 0 - ^-s) < 0.

B. Let M be the manifold R + x S with the Lorentz metric

2 2 a 2 ds = dr - r dj ,a £ F x {0}. The equation a a-1 ^ V r = ? Pr I 0 o i »r0 <°> ■ -O has the solution

„r (s) . {

1

(r„2‘a + ^ a(Z-a)s)7 ^ a t 2

roe a = 2

2r For a < 0 the solution exists for s e 1“ = (*,

-2-a

-2r 2-a a(2-a)

•» 0 as s + ■ and lim p (s) = ® . a(2-a) s--<» ro

and

(For a = 0 the map (e,s) -*• (r ,e) is a solution of the heat equation)

o 2r 2-a

For 0 < a < 2 the solution exists for s e Ia = (----2--- ,«) and

p (s ) + “> as s p (s) ♦ 0 as s

s o

-2r.2-a

a(2-a)

For a = 2 the solution exists for all values of s and lim p (s) = 0, s-*-oo ro

lim p (s) =_r s-»» o

For a > 2 the solution exists for s € I = (-=°,

-2r 2-a a(2-a) ) and lim p (s) = 0 and p (s) -*• as s -2r 2-a a(2-al

C. Let M be the manifold F x with the metric given by ds2 = dr2 + hdif2 where h = ±(2 + cosr). Suppose rQ e (-tt,0) U (0,tt). The equations

3s % = ± ? Sin % j

Pr (°) ■ r0 1

0

have the solutions p* (si = 2 arc tan [e4 ^ tan r /2] which exist for all values of the deformation parameter s.

For r e (-tt,0) pr (s) -► - tt and p* (s) + 0 as s + - » 0 0 p" (s) -*• 0 and p+ (s) + - ïï as s + “ For rQ £ (0,tt), pf (s) n and pr (s) + 0 as s + -o o + P- (s) -*■ 0 and p (s) ♦ i as s - °° o o

The curves S1 -*■ M, 8 - ( - i r , e ) , 0- ( O ,0), 0 - (tt,0) are geodesics for both metrics.